In many nonlinear systems, such as chaotic vibrations in structures or lasers, there exist regions in which much of the energy is present within a small range of frequencies. Typically, this occurs when the system is operating at or near resonance. Systems, such as aluminum wings or combustion engines, which operate at resonance, may fail due to repetitive stress caused by driving such a system near resonance.
Although numerous areas in science are now known to exhibit chaos as a natural occurrence, many situations would benefit from the inducement of chaos. In biology, the disappearance of chaos may signal pathological phenomena. In mechanics, chaos could be induced in order to prevent resonance, such as with the aluminum wings or combustion engines noted above. For example, in a system of coupled pendulums, one can excite chaotic motion of several modes to spread the energy over a wide frequency range. In optics, material damage is caused by lasers having a peak intensity at a given temporal frequency, so chaos is desirable since it has broadband spectra. It has also been suggested that chaos occur for normal machine tool cutting, making chaos preservation a desired control for deeper than normal cutting.
A conventional method maintains chaos in a regime where only chaotic transients exist, based on accurate analytical knowledge of the dynamical system, and requiring not only a priori phase space knowledge of escape regions from chaos, but also of preiterates of these regions. Such conventional methods maintain chaos using time series which require monitoring and adjusting the system prior to entering an escape region of the attractor by several iterates, an overly burdensome process.
Moreover, for the flow considered herein, preimages of sets in the escape regions cover much of the chaotic transient region, so monitoring all preimages, as done conventionally, would be tedious.